\(\int (e x)^{-1-2 n} (a+b (c+d x^n)^2)^p \, dx\) [345]

Optimal result

Integrand size = 25, antiderivative size = 210 \[ \int (e x)^{-1-2 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=-\frac {(e x)^{-2 n} \left (-\frac {x^{-n} \left (\sqrt {-a}-\sqrt {b} c-\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (\frac {x^{-n} \left (\sqrt {-a}+\sqrt {b} c+\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (a+b c^2+2 b c d x^n+b d^2 x^{2 n}\right )^p \operatorname {AppellF1}\left (2 (1-p),-p,-p,3-2 p,\frac {\left (\frac {\sqrt {-a}}{\sqrt {b}}-c\right ) x^{-n}}{d},-\frac {\left (\frac {\sqrt {-a}}{\sqrt {b}}+c\right ) x^{-n}}{d}\right )}{2 e n (1-p)} \] Output:

-1/2*(a+b*c^2+2*b*c*d*x^n+b*d^2*x^(2*n))^p*AppellF1(2-2*p,-p,-p,3-2*p,((-a 
)^(1/2)/b^(1/2)-c)/d/(x^n),-((-a)^(1/2)/b^(1/2)+c)/d/(x^n))/e/n/(1-p)/((e* 
x)^(2*n))/((-((-a)^(1/2)-b^(1/2)*c-b^(1/2)*d*x^n)/b^(1/2)/d/(x^n))^p)/(((( 
-a)^(1/2)+b^(1/2)*c+b^(1/2)*d*x^n)/b^(1/2)/d/(x^n))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 1.93 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.88 \[ \int (e x)^{-1-2 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=\frac {(e x)^{-2 n} \left (1+\frac {c x^{-n}}{d}+\frac {a x^{-n}}{\sqrt {-a b d^2}}\right )^{-p} \left (\frac {x^{-n} \left (\sqrt {-a b d^2}+b d \left (c+d x^n\right )\right )}{b d^2}\right )^{-p} \left (a+b \left (c+d x^n\right )^2\right )^p \operatorname {AppellF1}\left (2-2 p,-p,-p,3-2 p,-\frac {\left (b c d+\sqrt {-a b d^2}\right ) x^{-n}}{b d^2},\frac {\left (-b c d+\sqrt {-a b d^2}\right ) x^{-n}}{b d^2}\right )}{2 e n (-1+p)} \] Input:

Integrate[(e*x)^(-1 - 2*n)*(a + b*(c + d*x^n)^2)^p,x]
 

Output:

((a + b*(c + d*x^n)^2)^p*AppellF1[2 - 2*p, -p, -p, 3 - 2*p, -((b*c*d + Sqr 
t[-(a*b*d^2)])/(b*d^2*x^n)), (-(b*c*d) + Sqrt[-(a*b*d^2)])/(b*d^2*x^n)])/( 
2*e*n*(-1 + p)*(e*x)^(2*n)*(1 + c/(d*x^n) + a/(Sqrt[-(a*b*d^2)]*x^n))^p*(( 
Sqrt[-(a*b*d^2)] + b*d*(c + d*x^n))/(b*d^2*x^n))^p)
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2086, 1694, 1693, 1178, 150}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (e x)^{-2 n-1} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx\)

\(\Big \downarrow \) 2086

\(\displaystyle \int (e x)^{-2 n-1} \left (a+b c^2+2 b c d x^n+b d^2 x^{2 n}\right )^pdx\)

\(\Big \downarrow \) 1694

\(\displaystyle \frac {x^{2 n} (e x)^{-2 n} \int x^{-2 n-1} \left (2 b c d x^n+b d^2 x^{2 n}+b c^2+a\right )^pdx}{e}\)

\(\Big \downarrow \) 1693

\(\displaystyle \frac {x^{2 n} (e x)^{-2 n} \int x^{-3 n} \left (2 b c d x^n+b d^2 x^{2 n}+b c^2+a\right )^pdx^n}{e n}\)

\(\Big \downarrow \) 1178

\(\displaystyle -\frac {x^{2 n} (e x)^{-2 n} \left (x^{-n}\right )^{2 p} \left (-\frac {x^{-n} \left (\sqrt {-a}-\sqrt {b} c-\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (\frac {x^{-n} \left (\sqrt {-a}+\sqrt {b} c+\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (a+b c^2+2 b c d x^n+b d^2 x^{2 n}\right )^p \int \left (x^{-n}\right )^{1-2 p} \left (1-\frac {\left (\frac {\sqrt {-a}}{\sqrt {b}}-c\right ) x^{-n}}{d}\right )^p \left (\frac {\left (c+\frac {\sqrt {-a}}{\sqrt {b}}\right ) x^{-n}}{d}+1\right )^pdx^{-n}}{e n}\)

\(\Big \downarrow \) 150

\(\displaystyle -\frac {(e x)^{-2 n} \left (-\frac {x^{-n} \left (\sqrt {-a}-\sqrt {b} c-\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (\frac {x^{-n} \left (\sqrt {-a}+\sqrt {b} c+\sqrt {b} d x^n\right )}{\sqrt {b} d}\right )^{-p} \left (a+b c^2+2 b c d x^n+b d^2 x^{2 n}\right )^p \operatorname {AppellF1}\left (2-2 p,-p,-p,3-2 p,\frac {\left (\frac {\sqrt {-a}}{\sqrt {b}}-c\right ) x^{-n}}{d},-\frac {\left (c+\frac {\sqrt {-a}}{\sqrt {b}}\right ) x^{-n}}{d}\right )}{2 e n (1-p)}\)

Input:

Int[(e*x)^(-1 - 2*n)*(a + b*(c + d*x^n)^2)^p,x]
 

Output:

-1/2*((a + b*c^2 + 2*b*c*d*x^n + b*d^2*x^(2*n))^p*AppellF1[2 - 2*p, -p, -p 
, 3 - 2*p, (Sqrt[-a]/Sqrt[b] - c)/(d*x^n), -((Sqrt[-a]/Sqrt[b] + c)/(d*x^n 
))])/(e*n*(1 - p)*(e*x)^(2*n)*(-((Sqrt[-a] - Sqrt[b]*c - Sqrt[b]*d*x^n)/(S 
qrt[b]*d*x^n)))^p*((Sqrt[-a] + Sqrt[b]*c + Sqrt[b]*d*x^n)/(Sqrt[b]*d*x^n)) 
^p)
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1178

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(1/(d + e*x))^(2*p))*((a + 
b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q + 2*c* 
x)/(2*c*(d + e*x))))^p))   Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b 
 - q)/(2*c)))*x, x]^p*Simp[1 - (d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d 
 + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[m, 0]
 

rule 1693

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol 
] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, 
x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && IntegerQ 
[Simplify[(m + 1)/n]]
 

rule 1694

Int[((d_)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x 
_Symbol] :> Simp[d^IntPart[m]*((d*x)^FracPart[m]/x^FracPart[m])   Int[x^m*( 
a + b*x^n + c*x^(2*n))^p, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[ 
n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2086

Int[(u_)^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, 
x]^p, x] /; FreeQ[{d, m, p}, x] && TrinomialQ[u, x] &&  !TrinomialMatchQ[u, 
 x]
 

Maple [F]

\[\int \left (e x \right )^{-1-2 n} {\left (a +b \left (c +d \,x^{n}\right )^{2}\right )}^{p}d x\]

Input:

int((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^2)^p,x)
 

Output:

int((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^2)^p,x)
 

Fricas [F]

\[ \int (e x)^{-1-2 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{2} b + a\right )}^{p} \left (e x\right )^{-2 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^2)^p,x, algorithm="fricas")
 

Output:

integral((b*d^2*x^(2*n) + 2*b*c*d*x^n + b*c^2 + a)^p*(e*x)^(-2*n - 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^{-1-2 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=\text {Timed out} \] Input:

integrate((e*x)**(-1-2*n)*(a+b*(c+d*x**n)**2)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^{-1-2 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{2} b + a\right )}^{p} \left (e x\right )^{-2 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^2)^p,x, algorithm="maxima")
 

Output:

integrate(((d*x^n + c)^2*b + a)^p*(e*x)^(-2*n - 1), x)
 

Giac [F]

\[ \int (e x)^{-1-2 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=\int { {\left ({\left (d x^{n} + c\right )}^{2} b + a\right )}^{p} \left (e x\right )^{-2 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^2)^p,x, algorithm="giac")
 

Output:

integrate(((d*x^n + c)^2*b + a)^p*(e*x)^(-2*n - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1-2 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=\int \frac {{\left (a+b\,{\left (c+d\,x^n\right )}^2\right )}^p}{{\left (e\,x\right )}^{2\,n+1}} \,d x \] Input:

int((a + b*(c + d*x^n)^2)^p/(e*x)^(2*n + 1),x)
 

Output:

int((a + b*(c + d*x^n)^2)^p/(e*x)^(2*n + 1), x)
 

Reduce [F]

\[ \int (e x)^{-1-2 n} \left (a+b \left (c+d x^n\right )^2\right )^p \, dx=\text {too large to display} \] Input:

int((e*x)^(-1-2*n)*(a+b*(c+d*x^n)^2)^p,x)
 

Output:

( - (x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)**p + 2*x**(2*n)*int((x** 
(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)**p/(x**(3*n)*a*b*d**2*p*x - 2*x* 
*(3*n)*a*b*d**2*x + x**(3*n)*b**2*c**2*d**2*p*x - 2*x**(3*n)*b**2*c**2*d** 
2*x + 2*x**(2*n)*a*b*c*d*p*x - 4*x**(2*n)*a*b*c*d*x + 2*x**(2*n)*b**2*c**3 
*d*p*x - 4*x**(2*n)*b**2*c**3*d*x + x**n*a**2*p*x - 2*x**n*a**2*x + 2*x**n 
*a*b*c**2*p*x - 4*x**n*a*b*c**2*x + x**n*b**2*c**4*p*x - 2*x**n*b**2*c**4* 
x),x)*a*b*c*d*n*p**2 - 4*x**(2*n)*int((x**(2*n)*b*d**2 + 2*x**n*b*c*d + a 
+ b*c**2)**p/(x**(3*n)*a*b*d**2*p*x - 2*x**(3*n)*a*b*d**2*x + x**(3*n)*b** 
2*c**2*d**2*p*x - 2*x**(3*n)*b**2*c**2*d**2*x + 2*x**(2*n)*a*b*c*d*p*x - 4 
*x**(2*n)*a*b*c*d*x + 2*x**(2*n)*b**2*c**3*d*p*x - 4*x**(2*n)*b**2*c**3*d* 
x + x**n*a**2*p*x - 2*x**n*a**2*x + 2*x**n*a*b*c**2*p*x - 4*x**n*a*b*c**2* 
x + x**n*b**2*c**4*p*x - 2*x**n*b**2*c**4*x),x)*a*b*c*d*n*p + 2*x**(2*n)*i 
nt((x**(2*n)*b*d**2 + 2*x**n*b*c*d + a + b*c**2)**p/(x**(3*n)*a*b*d**2*p*x 
 - 2*x**(3*n)*a*b*d**2*x + x**(3*n)*b**2*c**2*d**2*p*x - 2*x**(3*n)*b**2*c 
**2*d**2*x + 2*x**(2*n)*a*b*c*d*p*x - 4*x**(2*n)*a*b*c*d*x + 2*x**(2*n)*b* 
*2*c**3*d*p*x - 4*x**(2*n)*b**2*c**3*d*x + x**n*a**2*p*x - 2*x**n*a**2*x + 
 2*x**n*a*b*c**2*p*x - 4*x**n*a*b*c**2*x + x**n*b**2*c**4*p*x - 2*x**n*b** 
2*c**4*x),x)*b**2*c**3*d*n*p**2 - 4*x**(2*n)*int((x**(2*n)*b*d**2 + 2*x**n 
*b*c*d + a + b*c**2)**p/(x**(3*n)*a*b*d**2*p*x - 2*x**(3*n)*a*b*d**2*x + x 
**(3*n)*b**2*c**2*d**2*p*x - 2*x**(3*n)*b**2*c**2*d**2*x + 2*x**(2*n)*a...